Mean spatial gradient

The data (Figure 12) show that the mean spatial gradients of nitrate, phosphate, and silicate are spatially coherent and in phase with each other. These gradients, however, are out of phase by about 180° with the corre-... 

Figure 12. Mean spatial gradients fv) of the various water properties sampled along inshore segments. The letters correspond to the cross-shelf transects in Figure 5b.

The form of (1.15) is identical to the balance equation that is used in finite-volume CFD codes for passive scalar mixing.17 The principal difference between a zone model and a finite-volume CFD model is that in a zone model the grid can be chosen to optimize the capture of inhomogeneities in the scalar fields independent of the mean velocity and turbulence fields.18 Theoretically, this fact could be exploited to reduce the number of zones to the minimum required to resolve spatial gradients in the scalar fields, thereby greatly reducing the computational requirements. 

The CFD code must use a grid that also resolves spatial gradients in the mean velocity and turbulence fields. At some locations in the reactor, the scalar fields may be constant, and thus a coarser grid (e.g., a zone) can be employed. 

More precisely, the spatial gradient of the mean velocity is independent of position in a homogeneous turbulent shear flow. 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

Since the liquid is perfectly mixed, the density is the same everywhere in the tank it does not vary with radial or axial position i.e., there are no spatial gradients in density in the tank. This is why we can use a macroscopic system. It also means that there is only one independent variable, t. 

The second model, the so-called gradient-flux law, is considered to be more fundamental, although it is based on a more restrictive physical picture. In contrast to the mass transfer model, in which no assumption is made regarding the spatial separation of subsystems A and B, in the gradient-flux law it is assumed that the subsystems and the distance between them, Axa/b, become infinitely small. For very small subsystems the term occupation number loses its meaning and must be replaced by occupation density or concentration. Obviously, the difference in occupation density tends toward zero, as well. Yet the ratio of the two differences, Aoccupa-tion density Axa/b, is equal to the spatial gradient of the occupation density and usually different from zero ... 

If the boundary conditions (i.e., the aldehyde concentration in the atmosphere, [A]a, and in the interior of the water body, [A]w) are given and held constant, steady-state conditions are quickly established in the film 3[A]/3t = 3[D]/3t = 0. Since we assume that the diol cannot escape into the atmosphere, the slope of the [D]-profile must be zero at the water surface. Note that any spatial gradient at z= 0 would mean transport by molecular diffusion from or to the boundary. 

For no-flux boundary conditions, the spatial gradient at the boundary must have zero component normal to the boundary.49 In a circle of radius ro, this means that dx(r, o, t)ldr = dy(r, o, t)/dr = 0 at r = r0- The zeros in the derivatives of J (z) occur at particular values of the argument z = z. 50 Therefore, the spatial mode J k r) cos m >, which we abbreviate by J j, is obtained when the jth zero in the derivative of J occurs at the boundary that is, when k ro = z. This fixes the value of knJ associated with the mode J for any given radius ro. As the radius changes, the value of k j changes in inverse proportion. 

In practice, the fluid velocity profile is rarely flat, and spatial gradients of concentration and temperature do exist, especially in large-diameter reactors. Hence, the plug-flow reactor model (Fig. 7.1) does not describe exactly the conditions in industrial reactors. However, it provides a convenient mathematical means to estimate the performance of some reactors. As will be discussed below, it also provides a measure of the most efficient flow reactor—one where no mixing takes place in the reactor. The plug-flow model adequately describes the reactor operation when one of the following two conditions is satisfied ... 

Seasonal Cycles and Mean Annual Spatial Gradients... 

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